Let be a C-algebra, let Then in if and only if there exists a unitary element homotopic to the identity which implements unitary equivalence,
$q=upu^{}$.
Proof:
Throughout this proof, denotes the unit of .
Suppose first that for some . Let be a continuous map of unitaries in from to .
Then, since is an ideal of is a continuous path of projections in from to .(what?)
(It is clear that for each t, we have a projection, and the path is the desired one, continuity in P(A) requires A to be an ideal iI guess.)
Conversely, if , then there exists projections in such that for .(How on earth is it the case that these exist?)
It therefore suffices to prove the implication in the case where .
Put then , and Hence is invertible and in by Proposition 2.1.11.
Let be the polar decomposition for , see paragraph 2.19.
Then . From the properties of the polar decomposition we have that inside of , and this entails that belongs to . By Proposition 2.1.8.